Re: Nuber Theory and Sequences

Is the third line a part of the definition of U, just like the first two lines? If not, then it should be indicated.

There is no constant K that works for all n and p. For example, for p = 2, K has to be 1 for n = 1, and K has to be 2 for n = 2.

- Yes the third part is a part of the definition, and is true for all n and p that are greater than 2
- K is not a constant, i mean i could show that U(p) is the remainder of the division of U(n+p) on U(n).

Re: Nuber Theory and Sequences

- Yes the third part is a part of the definition, and is true for all n and p that are greater than 2

That's not the response I was hoping for. Look at the first two lines. The first line defines U at 1 and 2. The second line defines U at 3 when n = 2. Indeed, the right-hand side refers to U(2) and U(1), which are already defined. Similarly, it defined U at 4 when n = 3, and in general it defines U at every n ≥ 2. So, what can the third line add? At best it agrees with what the first two lines say; at worst it contradicts them. But it is not a part of the definition; it is a separate claim. It turns out that this claim is true and is given as a lemma without a proof to be used in this problem.

Not knowing the definitions involved in a statement is worse than not being able to prove that statement, but here are further hints anyway. Replace U(n+p) with U(n) * U(p-1) + U(n+1) * U(p) in GCD(U(n+p), U(n)) and use the following facts.

Re: Nuber Theory and Sequences

Originally Posted by emakarov

That's not the response I was hoping for. Look at the first two lines. The first line defines U at 1 and 2. The second line defines U at 3 when n = 2. Indeed, the right-hand side refers to U(2) and U(1), which are already defined. Similarly, it defined U at 4 when n = 3, and in general it defines U at every n ≥ 2. So, what can the third line add? At best it agrees with what the first two lines say; at worst it contradicts them. But it is not a part of the definition; it is a separate claim. It turns out that this claim is true and is given as a lemma without a proof to be used in this problem.

Not knowing the definitions involved in a statement is worse than not being able to prove that statement, but here are further hints anyway. Replace U(n+p) with U(n) * U(p-1) + U(n+1) * U(p) in GCD(U(n+p), U(n)) and use the following facts.

the 3rd line is the previous question in this exercice and I already solved it, I tought it may be used in this question that's why i put there.
as for GCD(U(n+1), U(n)) = 1 I proved it with induction.

Re: Nuber Theory and Sequences

Originally Posted by Orpheus

the 3rd line is the previous question in this exercice and I already solved it, I tought it may be used in this question that's why i put there.
as for GCD(U(n+1), U(n)) = 1 I proved it with induction.

Good, then proving GCD(U(n+p), U(n)) = GCD(U(n), U(p)) should not be hard.